In some region, the gravitational field is zero. The gravitational potential in this region

The correct option is: Must be constant

Let's understand the relation between the gravitational field and gravitational potential step-by-step.

The gravitational field vector $E$ at any point is defined as the negative gradient of the gravitational potential $V$ at that point:
$E=-\nabla V$
In one-dimensional motion (or along any direction $r$), this relationship can be simplified to:
$E=-\frac{dV}{dr}$

According to the problem statement, the gravitational field in the given region is zero. Therefore:
$E=0$
Substituting this value into the relation:
$-\frac{dV}{dr}=0$
⇒ $\frac{dV}{dr}=0$

In calculus, the derivative of a function with respect to a variable is zero if and only if the function is a constant.
Therefore:
$V=\text{constant}$

This mathematical deduction tells us that if the gravitational field is zero throughout a region, the gravitational potential in that region does not change with position; it must be constant. Note that it does not necessarily have to be zero, but it must be uniform (constant) throughout that region.